What Is a Turnpike?

Lionel W. McKenzie of the University of Rochester, a significant player in Knowledge and the Wealth of Nations and careful reader of the book, has along sent “a little paper” untangling the highly-compressed description of turnpike theorems that appears in connection with the series of conferences on growth theory that occurred in the early 1960s. [P. 156: “Exciting new theorems about the existence of “turnpikes” were being proved and disproved (routes by which economies might swiftly move to higher levels of industrial development through forced investment in heavy industy).” ] He writes:

I was surprised when I worked on the notion of a turnpike in economics that it was not already explored by mathematicians as a part of the calculus of variations. However I have found many topics in the mathematics of economics which were not dealt with in the mathematical literature not because they were difficult but because they were too special for general mathematical interest. However I have now noticed that some mathematicians have taken an interest in the general problem as a problem in calculus of variations and indeed have even cited my papers. The general problem of calculus of variations arises when we are presented with a set of functions F and a valuation function v which gives a value to every function in this set and we ask which function f * in the set F satisfies the condition that v(f*) (> or =) v(f) for any f in F. I believe the subject arose from considering in what curve a rope of given length suspended between two points would assume. This problem was transformed into the question of what curve would minimize the potential energy of the rope arising from the force of gravitation. The answer is a catenary.

The problem of optimal growth is concerned with maximizing the utility achieved by consumers given initial conditions and the production possibilities. To simplify the problem many researches treated the production possibilities in terms of periods of given length so that initial conditions were transformed over the period into terminal conditions, which would be the initial conditions of the subsequent period, and a certain amount of consumption during the period. The consumption would provide a certain amount of utility to consumers. The objective is to maximize the sum of utility over the future which may be taken to be infinite. The problem is to choose the levels of output in each period to achieve this maximum. If the future is infinite the sum may still be rendered finite if future utilities are discounted. However even if the sum is infinite an overtaking criterion may prove usable where the objective is to find a sequence of consumptions for which there is no other feasible sequence whose finite sums eventually permanently exceed the finite sums of the chosen path.

The turnpike theorem in this setting says that initial conditions, that is, initial capital stocks, do not matter in the long run but eventually all optimal paths converge. The proof of this proposition depends on the concavity of the utility function for each period as a function of the initial and terminal stocks of the period. This concavity in turn follows from the concavity of the utility functions of each consumer and the convexity of the production set. That is to say, if a certain input of capital stocks and output of capital stocks and consumption goods is possible and another such combination is also possible, then any intermediate combination is possible. This would be true for the Romer model if the set of designs were fixed. It remains true in the Lucas model if human capital is treated like other capital goods. Thus making technological change endogenous does not necessarily invalidate the turnpike theorems.

It is really quite easy to see that a turnpike theorem must hold when utility is not discounted (as in Ramsey) and the utility function is a strictly concave function of the initial and terminal capital stocks each period. A strictly concave function has the property that the value of the function halfway in between two values of the argument exceeds one half of the sum of the values of the function at the two values of the argument. Now assume that by sacrificing utility in the early periods it is possible to move from the second optimal path to the path lying halfway between the two presumed optimal paths and the paths do not converge. Then the utility sum along the intermediate path must eventually exceed the sum of utility along the second path and that path cannot be optimal, by the overtaking criterion. Therefore if both paths are optimal they must converge.

Of course, in Romer’s model this concavity of the utility function which would follow from convexity of the periodwise production set and concavity of the utility function (presumably the sum of utility functions belonging to the individual consumers) does not follow because the periodwise production sets are not convex. They are not convex because the inputs are not confined to physical goods and labor which would be rival and excludable. The inputs include a list of designs or instructions on how to do things which are nonrival and at least partially excludable. Indeed it is not clear how the notion of convexity could be applied to the list of designs. On the other hand, if in the manner of Lucas you deal with technological change by treating it as a product of a research industry with inputs of human capital and ordinary goods it is possible to think of these industries as having convex production sets where using half the inputs of one process and half the inputs of another would allow the output of more than half the output of designs from one process and half the output of designs from the other. Of course there would be a measurement problem but this is already present with the physical goods where valuations may be used.

Romer’s basic argument against the strict convexity of the production set, that is, the set of inputs and outputs that are possible, is found on page S76 of his 1990 article in the JPE. It is equivalent to consider the set of outputs as a convex function of the set of inputs which he writes F(A,X) where A is the list of designs and X is the set of goods and services, including capital goods and labor services. He asserts that F is indeed concave in X while as far as I can see concavity is not given a meaning for the set of designs, but an increase in the number of designs can increase output. Clearly no problem for the turnpike argument arises if the list of designs does not change, so that the argument proceeds purely in terms of X. Equally I would suppose no problem arises for the turnpike if the view is taken that a new design does not contributes to production until it is embodied in physical goods. Of course, turnpikes are not the major interest of Romer, who wants to explain what underlies the differences in the wealth of nations. Another difficulty arises with the turnpike theorem applied to one country, since one country receives inputs from other countries and supplies outputs to them. If these opportunities remain constant, it is well and good. But if they change with circumstances, the appropriate region for applying the turnpike theorem would seem to the entire collection of countries that trade

Of course, no model is going to represent the real world with accuracy. One merely seeks models that fit at least loosely. Indeed, most of the literature on convergence of countries to similar levels of wealth per capita are macro models, that is, they are aggregated to a maximum extent and cannot possibly represent the real world in any but a very loose sense. The usual setting of the turnpike is a disaggregated model and the convergence is to a complex of capital goods. The message is that with a given technology and a given population along an optimal path the eventual relative composition of the stock of capital goods, including perhaps human capital, does not depend on the initial composition of capital goods, but only on the production functions.

The final theorems on this subject in my book (Classical General Equilibrium Theory, MIT Prerss, 2002) point out that a competitive equilibrium over time with perfect foresight does achieve an optimal path in terms of a welfare function that aggregates the utilities of the consumers in proportion to their respective wealths. Incidentally the French economist Maurice Allais argued that every country was saturated in capital goods given its culture and social structure. I always found that an intriguing idea. A final note, the turnpike for population growth is well known in the theory of demography. That the equilibrium distribution of a population over different ages does not depend on the initial distribution of population over but only on the fertility of the women at different ages. This was first embodied in the writing of Ansley Coale, one of the greatest demographers of recent times, who happened to be a graduate student in economics with me at Princeton in 1939-42.

Both Barro and Romer did much of their best work when on the Rochester faculty. The same is true of Sherwin Rosen (and Bob Fogel who won a Nobel prize. I believe Sherwin was cheated of his prize by cancer.) I call Romer my grandstudent, since his supervisor was Jose Scheinkman whose PhD was supervised by me.

The idea of a turnpike theorem is due to Paul Samuelson when he wrote a paper for the Rand Corporation around 1950. His objective was to give economic meaning to the model of John von Neumann published around 1936, in the reports of the symposium run in Vienna by the mathematician Karl Menger, son of the economist Carl Menger. Incidentally the first proofs of existence of competitive equilibrium were published by me and by Arrow and Debreu in 1954, mine a few months before theirs. The proofs were completely independently done. They were both given at the Chicago meeting of the Econometric Society in 1952. I mentioned their paper, but they did not mention mine, apparently because Debreu, who was present at my presentation, never told Arrow about my paper. This was told to me by Arrow. My immediate objective was to prove the existence and uniqueness of the solution for Graham’s model of world trade, which his students solved by trial and error, but it was applied to the general case of a competitive economy. Graham asked von Neumann to find a solution to his model analytically and von Neumann told him it was impossible, which of course was true. Herbert Scarf finally found an iterative procedure that converges to a competitive general equilibrium, and later versions of this procedure are used in Washington perhaps more often than the Leontief and Klein models. Of course, Scarf deserves a Nobel Prize for this accomplishment (which Leontief and Klein got) but I guess it is no disgrace to join T. S. Eliot, James Joyce, and Marcel Proust among those who did not receive a Nobel prize for their work.

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